JournalsprimsVol. 49, No. 2pp. 313–360

Transcendental Kähler Cohomology Classes

  • Dan Popovici

    Université Paul Sabatier, Toulouse, France
Transcendental Kähler Cohomology Classes cover
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Associated with a real, smooth, dd-closed (1,1)(1, \, 1)-form α\alpha of possibly non-rational De Rham cohomology class on a compact complex manifold XX is a sequence of asymptotically holomorphic complex line bundles LkL_k on XX equipped with (0,1)(0, \, 1)-connections ˉk\bar\partial_k for which ˉk20\bar\partial_k^2\neq 0. Their study was begun in the thesis of L. Laeng. We propose in this non-integrable context a substitute for H\"ormander's familiar L2L^2-estimates of the ˉ\bar\partial-equation of the integrable case that is based on analysing the spectra of the Laplace-Beltrami operators Δk\Delta_k'' associated with ˉk\bar\partial_k. Global approximately holomorphic peak sections of LkL_k are constructed as a counterpart to Tian's holomorphic peak sections of the integral-class case. Two applications are then obtained when α\alpha is strictly positive\!: a Kodaira-type approximately holomorphic projective embedding theorem and a Tian-type almost-isometry theorem for compact K\"ahler, possibly non-projective, manifolds. Unlike in similar results in the literature for symplectic forms of integral classes, the peculiarity of α\alpha lies in its transcendental class. This approach will be hopefully continued in future work by relaxing the positivity assumption on α\alpha.

Cite this article

Dan Popovici, Transcendental Kähler Cohomology Classes. Publ. Res. Inst. Math. Sci. 49 (2013), no. 2, pp. 313–360

DOI 10.4171/PRIMS/107